First, find the expression for \(gf(x)\):

\(gf(x) = g(f(x)) = g(x^2 - 2x) = 2(x^2 - 2x) + 3\).

Simplify the expression:

\(gf(x) = 2x^2 - 4x + 3\).

To find if there are real solutions, solve \(2x^2 - 4x + 3 = 0\).

Calculate the discriminant \(b^2 - 4ac\) for the quadratic equation \(ax^2 + bx + c = 0\):

Here, \(a = 2\), \(b = -4\), \(c = 3\).

\(b^2 - 4ac = (-4)^2 - 4 \times 2 \times 3 = 16 - 24 = -8\).

Since the discriminant is negative, the quadratic equation has no real solutions.

Therefore, the equation \(gf(x) = 0\) has no real solutions.